Differential and Integral Equations are powerful tools to model and analyze biological problems. In this talk, two different biological applications will be discussed: one is in biomedical images and the other is in cellular biology.

The basic medical science research and clinical diagnosis and treatment have strongly benefited from the development of various noninvasive biomedical imaging techniques and modeling, e.g. magnetic resonance imaging (MRI) and computed tomography (CT). We introduce integro-differential models to the morphology and connectome study of human brains from brain images, as well as the shape analysis of ciliary muscles from human eyes.

In the application of cellular biology, we investigate the cell differentiation model of T cells. T cells of the immune system, upon maturation, differentiate into either Th1 or Th2 cells that have different functions. The decision to which cell type to differentiate depends on the concentrations of transcription factors T-bet and GATA-3. We study a population density model of the T cells and show that, under some conditions on the parameters of the system of integro-differential equations, various T cells differentiation scenarios occur.